On Ill‐ and Well‐Posedness of Dissipative Martingale Solutions to Stochastic 3D Euler Equations

We are concerned with the question of well‐posedness of stochastic, three‐dimensional, incompressible Euler equations. In particular, we introduce a novel class of dissipative solutions and show that (i) existence; (ii) weak–strong uniqueness; (iii) nonuniqueness in law; (iv) existence of a strong M...

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Published inCommunications on pure and applied mathematics Vol. 75; no. 11; pp. 2446 - 2510
Main Authors Hofmanová, Martina, Zhu, Rongchan, Zhu, Xiangchan
Format Journal Article
LanguageEnglish
Published Melbourne John Wiley & Sons Australia, Ltd 01.11.2022
John Wiley and Sons, Limited
Subjects
Dissipation
Euler-Lagrange equation
Martingales
Mathematical analysis
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Summary:We are concerned with the question of well‐posedness of stochastic, three‐dimensional, incompressible Euler equations. In particular, we introduce a novel class of dissipative solutions and show that (i) existence; (ii) weak–strong uniqueness; (iii) nonuniqueness in law; (iv) existence of a strong Markov solution; (v) nonuniqueness of strong Markov solutions: all hold true within this class. Moreover, as a by‐product of (iii) we obtain existence and nonuniqueness of probabilistically strong and analytically weak solutions defined up to a stopping time and satisfying an energy inequality. © 2021 Wiley Periodicals LLC.
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ISSN:0010-3640
1097-0312
DOI:10.1002/cpa.22023